What is this vector notation?
- Thread starter egal
- Start date
topsquark
Senior Member
- Joined
- Aug 27, 2012
- Messages
- 2,269
I'm not sure how prevalent this notation is in "ordinary" Mathematics but the notation |V> is a vector called a "bra" in Quantum Physics. It is simply an ordinary vector in an ordinary vector space. There is a corresponding vector <V| in the dual vector space to the set of all |V>'s and we call the inner product of these a "bra-ket" <V|V>, which audibly explains the words for the symbols.
So if we have a linear operator [math]\Omega[/math] and a scalar [math]\alpha[/math] then [math]\Omega ( \alpha |V> ) = \alpha ~ \Omega |V>[/math] and the rest follows.
I should mention that in Quantum Mechanics [math]\alpha[/math] and [math]\beta[/math] could very well be complex numbers, in which case
[math]( <V| \alpha ) \Omega = \alpha ^* <V| \Omega[/math] so the second line might not be correct.
-Dan
Addendum: Sorry. Most of the LaTeX should be written in a single sentence. I don't know why it did this or how to fix it. But it's readable.
So if we have a linear operator [math]\Omega[/math] and a scalar [math]\alpha[/math] then [math]\Omega ( \alpha |V> ) = \alpha ~ \Omega |V>[/math] and the rest follows.
I should mention that in Quantum Mechanics [math]\alpha[/math] and [math]\beta[/math] could very well be complex numbers, in which case
[math]( <V| \alpha ) \Omega = \alpha ^* <V| \Omega[/math] so the second line might not be correct.
-Dan
Addendum: Sorry. Most of the LaTeX should be written in a single sentence. I don't know why it did this or how to fix it. But it's readable.
Last edited:
@topsquark Use imath instead of math for inline math. Or, using the f(x) icon in the toolbar, make sure to select inline.
topsquark
Senior Member
- Joined
- Aug 27, 2012
- Messages
- 2,269
This is LaTeX test. No content has been changed.
I'm not sure how prevalent this notation is in "ordinary" Mathematics but the notation |V> is a vector called a "bra" in Quantum Physics. It is simply an ordinary vector in an ordinary vector space. There is a corresponding vector <V| in the dual vector space to the set of all |V>'s and we call the inner product of these a "bra-ket" <V|V>, which audibly explains the words for the symbols.
So if we have a linear operator [imath]\Omega[/imath] and a scalar [imath]\alpha[/imath] then [imath]\Omega ( \alpha |V> ) = \alpha ~ \Omega |V>[/imath] and the rest follows.
I should mention that in Quantum Mechanics [imath]\alpha[/imath] and [imath]\beta[/imath] could very well be complex numbers, in which case
[imath]( <V| \alpha ) \Omega = \alpha ^* <V| \Omega[/imath] so the second line might not be correct.
It works! Yay!
-Dan
I'm not sure how prevalent this notation is in "ordinary" Mathematics but the notation |V> is a vector called a "bra" in Quantum Physics. It is simply an ordinary vector in an ordinary vector space. There is a corresponding vector <V| in the dual vector space to the set of all |V>'s and we call the inner product of these a "bra-ket" <V|V>, which audibly explains the words for the symbols.
So if we have a linear operator [imath]\Omega[/imath] and a scalar [imath]\alpha[/imath] then [imath]\Omega ( \alpha |V> ) = \alpha ~ \Omega |V>[/imath] and the rest follows.
I should mention that in Quantum Mechanics [imath]\alpha[/imath] and [imath]\beta[/imath] could very well be complex numbers, in which case
[imath]( <V| \alpha ) \Omega = \alpha ^* <V| \Omega[/imath] so the second line might not be correct.
It works! Yay!
-Dan
I guess my math is not good enough to understand these, now I'm learning linear algebra then I'll look into these equations, thanks for your time
topsquark
Senior Member
- Joined
- Aug 27, 2012
- Messages
- 2,269
The definition of a linear operator is pretty basic to Linear Algebra. If you haven't seen that in your course yet then just be patient... you'll get there. On the other hand once you've seen it, if you are still having problems please make sure you ask lots of questions because it shows up everywhere.
-Dan
-Dan